Remark 1.1 Since the properties of operator \(\mathcal {L}\)  given in Proposition REF  are similar to those of the linearized Boltzmann collision operator and the operators \(\mathcal {Q}\)  and \(\mathcal {T}\)  has the estimates (REF ) and (REF ), we declare by the same arguments as Lemma A.1 and Lemma A.2 in [1]} that the initial remainder \(F_{R,\varepsilon }^\mathrm {in}(x,v)\)  in (REF ) can be chose so that \(F_\varepsilon ^\mathrm {in}(x,v) \geqslant 0\) . On the other hand, the fact \(c\leqslant 1-\mu \leqslant 1\)  for some positive constant \(c\)  implies \(F_\varepsilon ^\mathrm {in}(x,v) \leqslant 1\) .

